Type Grade Here - Wikispacesccgpsmathematicsk-5.wikispaces.com/file/view/K-Mat… · Web viewplease note that it is not until First Grade when “Understand the meaning of the equal

Georgia Standards of ExcellenceCurriculum Frameworks

GSE Kindergarten

Unit 5: Investigating Addition and Subtraction

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Mathematics

Georgia Department of EducationGeorgia Standards of Excellence Framework

GSE Investigating Addition and Subtraction Unit 5

Kindergarten Unit 5: Investigating Addition and Subtraction

TABLE OF CONTENTS (*indicates a new addition)

Overview..........................................................................................................................…3

Standards for Mathematical Practice and Content .............................................................5

Big Ideas..............................................................................................................................7

Essential Questions .............................................................................................................7

Concepts and Skills to Maintain .........................................................................................8

Strategies for Teaching and Learning..................................................................................8

Selected Terms and Symbols ..............................................................................................9

Common Misconceptions..................................................................................................10

Tasks..................................................................................................................................11

*Intervention Table............................................................................................................13

Addition and Subtraction Word Problems ............................................................14 Building Trains (5/10)! (Addition/Combining) ....................................................19 Building Trains (5/10)! (Subtraction/Separating)..................................................23 Going Bananas ......................................................................................................27 Tug-O-War ............................................................................................................34 Race to 20 Revisited .............................................................................................39 Shake and Spill (Addition/Combining) ................................................................42 Shake and Spill (Subtraction/Separating) .............................................................47 Sum-moning 5’s and 10’s .....................................................................................51 Decomposing Numbers Operations and Algebraic Thinking (FAL) ....................56 Bo Peep’s Domino Dilemma ................................................................................57 Bike Shop...............................................................................................................65 Calling Out Reds! 5/10..........................................................................................71 Dropping Pennies ..................................................................................................75 5/10 Go Fish ..........................................................................................................80 Young Farmer MacDonald ...................................................................................82 4 Sums in a Row ...................................................................................................88 Candy Store ...........................................................................................................93 Student Work Samples.........................................................................................100

***Please note that all changes made will appear in green. IF YOU HAVE NOT READ THE KINDERGARTEN CURRICULUM OVERVIEW IN ITS ENTIRETY PRIOR TO USE OF THIS UNIT, PLEASE STOP AND CLICK HERE: https://www.georgiastandards.org/Georgia-Standards/Frameworks/K-Math-Grade-Level-Overview.pdf Return to the use of this unit once you’ve completed reading the Curriculum Overview. Thank you.

Mathematics GSE Kindergarten Unit 5: Investigating Addition and SubtractionRichard Woods, State School Superintendent

July 2016 Page 2 of 105All Rights Reserved

https://www.georgiastandards.org/Georgia-Standards/Frameworks/K-Math-Grade-Level-Overview.pdf

https://www.georgiastandards.org/Georgia-Standards/Frameworks/K-Math-Grade-Level-Overview.pdf


GSE Investigating Addition and Subtraction Unit 5OVERVIEW

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

For numbers 0 – 10, Kindergarten students choose, combine, and apply strategies for answering quantitative questions. This includes quickly recognizing the cardinalities of less sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. Objects, pictures, actions, and explanations are used to solve problems and represent thinking. Although GSE states, “Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required”, please note that it is not until First Grade when “Understand the meaning of the equal sign” is an expectation.

For more information on the use of equations in Kindergarten please visit http://ccgpsmathematicsk-5.wikispaces.com/share/view/61530652

For more information on not using “take away” please visit http://ccgpsmathematicsk-5.wikispaces.com/share/view/61448708

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: join, add, separate, subtract, and, same amount as, equal, less, more, compose, decompose.

The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use to build their curriculum and to guide instruction.

(1)Representing, relating, and operating on whole numbers, initially with sets of objects.Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations, such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of less sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview





Number Sense Trajectory – Putting It All Together

Tra

ject

ory

SubitizingBeing able to visually recognize a quantity of 5 or less.

ComparisonBeing able to compare quantities by identifying which has more and which has less.

CountingRote procedure of counting. The meaning attached to counting is developed through one-to-one correspondence.

One-to-OneCorrespondenceStudents can connect one number with one object and then count them with understanding.

CardinalityTells how many things are in a set. When counting a set of objects, the last word in the counting sequence names the quantity for that set.

Hierarchical InclusionUnderstands numbers are nested inside of each other and that the number grows by one each count. 9 is inside 10 or 10 is the same as 9 + 1.

Number Conservation

The total number of objects remains the same when they are rearranged spatially. 5 is 4&1 OR 3&2.

Each concept builds on the previous idea and students should explore and construct concepts in such a sequence

Num

ber

Rel

atio

nshi

ps

Spatial RelationshipPatterned Set Recognition

Students can learn to recognize sets of objects in patterned arrangements and tell how many without counting.

One and Two-More or LessStudents need to understand the relationship of number as it relates to +/- one or two. Here students should begin to see that 5 is 1 more than 4 and that it is also 2 less than 7.

Understanding AnchorsStudents need to see the relationship between numbers and how they relate to 5s and 10s. 3 is 2 away from 5 and 7 away from 10.

Part-Part-Whole Relationship

Students begin to conceptualize a number as being made up from two or more parts.

Addition and Subtraction Strategies One/Two More/Less

These facts are a direct application of the One/Two More/ Less than relationships

Make a TenUse a quantity from one addend to give to another to make a ten then add the remainder.9 + 7 = 10 + 6

Near DoublesUsing the doubles anchor and combining it with 1 and 2 more/less.

Facts with ZeroNeed to be introduced so that students don’t overgeneralize that answers to addition are always greater.

DoublesMany times students will use doubles as an anchor when adding and subtracting.



Developing Number Sense: The Big PictureThis trajectory is designed to show number sense development through the early years of elementary school. Although the graphic organizer flows horizontally left to right, it also aligns vertically. Each concept builds from the previous stage and is the foundation to developing the number sense required of all students.

Developed by Graham Fletcher and adapted from:Clements& Sarama Learning and teaching early math: The learning trajectories approach (2009)Van de Walle & Lovin Teaching Student Centered Mathematics (k-3)



STANDARDS FOR MATHEMATICAL PRACTICE

The standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of the unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

Students are expected to:

1. Make sense of problems and persevere in solving them. Students are able to compose and decompose numbers while solving problems involving addition and subtraction. 2. Reason abstractly and quantitatively. Students begin to draw pictures, manipulate objects, use diagrams or charts, etc. to express quantitative ideas such as a joining situation or separating situations.3. Construct viable arguments and critique the reasoning of others. Students begin to clearly explain their thinking using mathematical language when composing and decomposing numbers.(Verbal and/or Written)4. Model with mathematics. Students will begin to apply their mathematical thinking to real- world situations when given an addition or subtraction word problem. 5. Use appropriate tools strategically. Students will use manipulatives such as counting bears, cubes and number lines to model addition and subtraction problems. 6. Attend to precision. Students attend to the language of real-world situations to make sense of addition and subtraction problems.7. Look for and make use of structure. Students begin to look for patterns and structure in the number system while exploring part-whole relationships using manipulatives.8. Look for and express regularity in repeated reasoning. Students begin to recognize and use multiple strategies when combining and decomposing sets of numbers.

(For descriptors of standard cluster please see the Grade Level Overview)***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

STANDARDS FOR MATHEMATICAL CONTENT

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking fromMGSEK.OA.1 Represent addition and subtraction with objects, fingers, mental images,

drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

MGSEK.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

MGSEK.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation. (drawings need not include an equation).





MGSEK.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

MGSEK.OA.5 Fluently add and subtract within 5.Problem Types

Result Unknown Change Unknown Start Unknown

Join/Combine

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?

2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?

? + 3 = 5

Separate/Decompose

Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown1

Put Together / Take Apart2

Three red apples and two green apples are on the table. How many apples are on the table?3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green?3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

Difference Unknown Greater Unknown Less UnknownCompare3 (“How many more?”

version):Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?(“How many fewer?” version):Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

(Version with “more”):Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?(Version with “fewer”):Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?2 + 3 = ?, 3 + 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?(Version with “fewer”):Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?5 – 3 = ?, ? + 3 = 5





2 + ? = 5, 5 – 2 = ?\6Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).

BIG IDEAS

Addition and subtraction problems are placed in four basic categories: Joining problems, Separating problems, Part-Part Whole problems, and Comparing problems.

A joining problem involves three quantities involves: the starting amount, the change amount, and the resulting amount.

A separating problem involves three quantities; the starting amount, the change amount (the amount being removed), and the resulting amount; however, the starting amount is the greatest amount with the change amount being removed which leaves the resulting amount.

Part-Part-Whole problems involve three quantities: two parts that are combined into one whole

Compare problems involve the comparison between two different quantities. The third quantity does not actually exist but is the difference between the two quantities. When one quantity is compared to another, the first quantity is either more than, less than, or equal to the second quantity.

Problems can be solved in different ways. Problems can be modeled using objects, pictures, and words. Various combinations of numbers can be used to represent the same quantity.

(see table on previous page for examples)

ESSENTIAL QUESTIONS

How can I represent and solve problem situations using objects, pictures, words and numbers?

What happens when I join quantities together? What happens when sets are joined or separated? How can I use different combinations of numbers to represent the same quantity? How can using benchmark numbers help me when adding or subtracting? How can I use models to represent addition and subtraction? How can using benchmark numbers help me when adding or subtracting? Why is it important that I can build the number combinations for the number 5? 10? What happens when I decompose a quantity? How do you know when your answer

makes sense? Does the order of addends change the sum? How do you know when your answer makes

sense?CONCEPTS/SKILLS TO MAINTAIN Although many students may have attended pre-school prior to entering kindergarten, this is the first year of school for some students. For that reason, no concepts/skills to maintain will be listed at this time. It is expected that teachers will differentiate to accommodate those students that may enter kindergarten with prior knowledge.





STRATEGIES FOR TEACHING AND LEARNING

Provide contextual situations for addition and subtraction that relate to the everyday lives of kindergarteners. A variety of situations can be found in children’s literature books. Students then model the addition and subtraction using a variety of representations such as drawings, sounds, acting out situations, verbal explanations and numerical expressions. Manipulatives, like two-color counters, clothespins on hangers, connecting cubes and stickers can also be used for modeling these operations. Kindergarten students should see addition and subtraction equations written by the teacher. Although students might struggle at first, teachers should encourage them to try writing the equations. Students’ writing of equations in Kindergarten is encouraged, but it is not required.

Create written addition or subtraction problems with sums and differences less than or equal to 10 using the numbers 0 to 10. It is important to use a problem context that is relevant to kindergarteners. After the teacher reads the problem, students choose their own method to model the problem and find a solution. Students discuss their solution strategies while the teacher represents the situation with an equation written under the problem. The equation should be written listing the numbers and symbols for the unknown quantities in the order that follows the meaning of the situation. The teacher and students should use the words equal and is the same as interchangeably.

Have students decompose numbers less than or equal to 5 during a variety of experiences to promote their fluency with sums and differences less than or equal to 5 that result from using the numbers 0 to 5. For example, ask students to use different models to decompose 5 and record their work with drawings or equations. Next, have students decompose 6, 7, 8, 9, and 10 in a similar fashion. As they come to understand the role and meaning of arithmetic operations in number systems, students gain computational fluency, using efficient and accurate methods for computing.

The teacher can use scaffolding to teach students who show a need for more help with counting. For instance, ask students to build a tower of 5 using 2 green and 3 blue linking cubes while you discuss composing and decomposing 5. Have them identify and compare other ways to make a tower of 5. Repeat the activity for towers of 7 and 9. Help students use counting as they explore ways to compose 7 and 9.

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.





Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience. Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context. Fluent students:

flexibly use a combination of deep understanding, number sense, and memorization. are fluent in the necessary baseline functions in mathematics so that they are able to

spend their thinking and processing time unpacking problems and making meaning from them.

are able to articulate their reasoning. find solutions through a number of different paths.

For more about fluency, see: http://MGSEK.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-tests.pdfSELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by students. Teachers should first present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or use them with words, models, pictures, or numbers.

Combine Compose Separate Decompose Compare Quantity

Common Core Georgia Performance Standards Mathematics Glossary

COMMON MISCONCEPTIONS

Students may over-generalize the vocabulary in word problems and think that certain words indicate solution strategies that must be used to find an answer. They might think that the word more always means to add and the words take away or left always means to subtract. When students use the words take away to refer to subtraction and its symbol, teachers need to repeat students’ ideas using the words minus or subtract. For example, students use addition to solve



http://www.corestandards.org/Math/Content/mathematics-glossary/glossary/

http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf



this Take from/Start Unknown problem: Seth took the 8 stickers he no longer wanted and gave them to Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin with?

If students progress from working with manipulatives to writing numerical expressions and equations, they skip using pictorial thinking. Students will then be more likely to use finger counting and rote memorization for work with addition and subtraction. Counting forward builds to the concept of addition while counting back leads to the concept of subtraction. However, counting is an inefficient strategy. Teachers need to provide instructional experiences so that students progress from the concrete level, to the pictorial level, then to the abstract level when learning mathematics.

TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all Kindergarteners. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning.

TASK DESCRIPTIONS

Scaffolding Task

Tasks that build up to the learning task.

Constructing Task

Constructing understanding through deep/rich contextualized problem solving tasks.

Practice Task Tasks that provide students opportunities to practice skills and concepts.Culminating Task

Designed to require students to use several concepts learned during the unit to answer a new or unique situation. Allows students to give evidence of their own understanding toward the mastery of the standard and requires them to extend their chain of mathematical reasoning.

Formative Assessment Lesson (FAL)

Lessons that support teachers in formative assessment which both reveal and develop students’ understanding of key mathematical ideas and applications. These lessons enable teachers and students to monitor in more detail their progress towards the targets of the standards.





Task Name Standards Task Type/Grouping Strategy Content Addressed Brief Description

Addition/Subtraction Word Problems

MGSEK.OA.1-5 Constructing TaskIndividual. Whole, Small

Group

Addition and subtraction through

word problems

Students answer different type addition and subtraction story

problems

Building Trains

MGSEK.OA.1,3,4,5Constructing Task

Whole or Small GroupNumber combinations

to 5 and 10

Students practice different ways of

building five using two different quantities

Going BananasMGSEK.OA.1-5 Practice Task

Whole, Small Group or Individual

Number combinations to 5 through problem

solving

Students find different combinations to build 5

and 10

Tug-O-WarMGSEK.OA.1,4 Practice Task

Partners

Using a number line to find number

combinations to 20

Students practice adding and subtracting to 10

Race to 20 Revisited

MGSEK.OA.1,5Practice Task

PartnersNumber combinations

to 5 (+/-)

Students practice finding the benchmark number 5 when adding

to 20

Shake and SpillAddition/Subtraction

MGSEK.OA.1,3,4,5 Constructing TaskWhole, Small Group or

Individual

Number combinations to 10 (+/-)

Students build number combinations to 10

Summoning 5’s and 10’s

MGSEK.OA.1,3,4,5Practice Task

Partners

Using 5 as a benchmark number, number

combinations to 10

Students practice using the benchmark number

5 to build number combinations to 10

Decomposing Numbers Operations and

Algebraic Thinking

MGSEK.OA.1-5

FAL

Using 5 and 10 as benchmarksAddition and Subtraction

Students show their understanding of

benchmark numbers in a formative assessment

lessonMathematics GSE Kindergarten Unit 5: Investigating Addition and Subtraction

Richard Woods, State School Superintendent July 2016 Page 11 of 105

All Rights Reserved

https://ccgpsmathematicsk-5.wikispaces.com/K-5+Formative+Assessment+Lessons+(FALs)





Bo Peep’s Domino Dilemma

MGSEK.OA.1-5 Practice TaskWhole, Small Group or

Individual

Number combinations to 5 and 10

Students use a variety of activities to build

combinations to 10

The Bike Store MGSEK.OA.1-5 Performance TaskIndividual


solving

Students use different combinations to five to

solve a problem.

Calling Out Reds

MGSEK.OA.1-5 Constructing TaskWhole, Small Group or

Individual

Mental computation of number combinations to

5 or 10.

Students practice building number

combinations to 10 mentally

Dropping PenniesMGSEK.OA.1-5 Performance Task

Individual


solving


combinations to 5

Make 5/10 Go Fish

MGSEK.OA.1-5Practice Task

Partners


5 or 10.


combinations to 5 and 10

Young Farmer MacDonald

MGSEK.OA.1-5 Practice TaskWhole, Small Group or

Individual


solving


combinations to 5

4 Sums in a Row

MGSEK.OA.1-5Practice Task

Partners


5 or 10.


combinations to 5 and 10

The Candy Store

MGSEK.OA.1-5Culminating Task

IndividualNumber combinations to 5 through problem

solving

Students use their knowledge gained to

build number combinations to 5

Each task is suggested but not required; teachers should choose the most appropriate tasks based on the needs of their students. For more information on these tasks in this unit please refer to the unit webinar found athttps://www.georgiastandards.org/Archives/Pages/default.aspx



https://www.georgiastandards.org/Archives/Pages/default.aspx



INTERVENTION TABLE

The Intervention Table below provides links to interventions specific to this unit. The interventions support students and teachers in filling foundational gaps revealed as students work through the unit. All listed interventions are from New Zealand’s Numeracy Project.

Cluster of Standards Name of Intervention

Snapshot of summary orStudent I can statement. . .

Materials Master

Understand addition as putting together and adding to, and understand subtraction as

taking apart and taking from

MGSEK.OA.1MGSEK.OA.2MGSEK.OA.3MGSEK.OA.4MGSEK.OA.5

Adding and Subtracting with

Counters

Solve addition problems to 20 by joining sets and counting all the objects

Fly Flip Solve addition problems to 20 by joining sets and counting all the objects MM 4-5

Both Hands

Solve addition problems to 20 by joining sets and counting all the objectsSolve subtraction problems from 20 by separating sets and counting all the objectsInstantly recognize patterns to 10 including doubles

Turtles 5 and… Solve addition problems to 20 by joining sets and counting all the objects MM 5-9

Adding and Subtracting with

One Hand

Solve subtraction problems from 20 by separating sets and counting all the objectsInstantly recognize patters to 5, including finger patterns

Teens and Fingers

Solve subtraction problems from 20 by separating sets and counting all the objectsRecall the facts up to 10, and the teen facts



http://nzmaths.co.nz/resource/teens-and-fingers

http://nzmaths.co.nz/resource/teens-and-fingers

http://nzmaths.co.nz/resource/adding-and-subtracting-one-hand



http://nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk5/MM%205_9.pdf

http://nzmaths.co.nz/resource/turtles-5-and

http://nzmaths.co.nz/resource/both-hands

http://nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk4/MM%204_5.pdf

http://nzmaths.co.nz/resource/fly-flip

http://nzmaths.co.nz/resource/adding-and-subtracting-counters









CONSTRUCTING TASK: Addition and Subtraction Word Problems Back To Task TableThis task can be modified and used throughout the unit

This task is intended to serve as a model for future tasks that you should create. “When children are learning about the operations of addition and subtraction, it’s helpful for them to see the connection between these processes and the world around them. Word problems accustom children to looking at groups of people or objects, help them see the actions of joining and separating, and give them experience figuring out sums and differences.” (Burns 2006) STANDARDS FOR MATHEMATICAL CONTENT

Understand addition as putting together and adding to, and understand subtractions as taking apart and taking from

MGSEK.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.




MGSEK.OA.5 Fluently add and subtract within 5.


1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.





BACKGROUND KNOWLEDGE

As you tell the addition or subtraction stories, have children act out the stories using a variety of manipulatives to represent the people, animals, or objects. Once children are comfortable acting out the stories and demonstrate proficiency with acting them out, show students how to record matching number sentences. The minus sign should be read as “minus” or “subtract” but not as “take away”. The plus sign is easier since it is typically a substitute for “and” (Van de Walle, 2010).

As you create word problems for your students, base them on people, animals, or objects that you are discussing in other content areas or that the students can connect with for other reasons. Learning is tied to emotion and if students can connect with the problem, they are more likely to be engaged and therefore, learn through their experience.

This task is designed to serve as an example of how you should recreate this type of task frequently throughout the unit and the remainder of the school year with a variety of books. Since there are several word problem types and you want to allow time for students to explain their thinking, this type of task can be spread across a few math segments to allow for rich mathematical discussion about each problem type.

For more information about common misconceptions, please refer to the unit overview.

ESSENTIAL QUESTIONS

How can I represent and solve problem situations using objects, pictures, words and numbers?

MATERIALS

Sample problem item sheet to use as a guide If You Give a Mouse a Cookie by Laura Numeroff or any book that can be used to create

word problems like the ones in the task chart Cookie shaped cereal for each student (or if using a different text, a variety of

manipulatives for students to choose from)

GROUPING

Whole, Individual and/or Small group





TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION:

Read the story, If You Give a Mouse a Cookie (or any book you choose) to students. Give students individual bags of the cookie shaped cereal to use as a math tool for this task. Explain to the students that you will work together to solve Mouse’s problem (you may choose to have students work in pairs or small groups during other math segments and later individually once they become more proficient with problem solving in a variety of situations). Discuss each problem on the chart and the various strategies used to find the sum (doubles, finding five or ten, drawing pictures, acting out the story with manipulatives, etc.).

Select word problems from the problem type sheet and change them as you see necessary to reflect the concepts you are teaching in other academic areas. Remember, they should only include numbers up to 10 .You could print a few to make a recording sheet so that students can represent their thinking on paper. You should make sure to expose students to a variety of structures (join problems, separate problems, part-part-whole problems, and compare problems) regularly so that they can develop proficiency with this type of problem-solving.

Also, do not expect that students will solve these problems in a single math segment. The emphasis is on the process, not the product. The goal is not only for a student to solve the problem, but rather for them to be able to explain their thinking through pictures, words, and numbers. Whatever they put on paper should explain what they did well enough to allow someone else to understand it (Van de Walle & Lovin 2006).

TEACHER REFLECTION QUESTIONS

Can students separate objects from a greater set of objects? Are students able to explain what happens when some objects are removed from a set of

objects? Added to a set of objects? Can students explain how to find out what is left when one quantity is removed from

another? Can students explain patterns as the story develops?

FORMATIVE ASSESSMENT QUESTIONS

What counting strategies did you use to separate (or combine) information? What is this story asking you to find out? What strategy will you use to solve this problem? Is there another way to solve this problem?

DIFFERENTIATION

Extension As students develop proficiency with solving each type of addition and subtraction

structure, have them write their own problems for others to solve. Teachers should encourage students to write various types of problems for others to solve.





Intervention

Allow students to work with numbers less than 5 so that they can practice using efficient strategies to solve the problems. Counting strategies are efficient at this stage, but will become inefficient and distracting as numbers become greater. As students begin to understand the relationships among numbers, they will begin learning number facts at a recall level (Carpenter et al. 1999).

Back to Intervention Table

ADDITIONAL RESOURCES

Carpenter, Thomas P., Fennema, Elizabeth, Loef Franke, Megan, Levi, Linda, Empson, Susan B., Children’s Mathematics: Cognitively Guided Math Instruction: Heinemann, 1999.

TECHNOLOGY

Early Math: Bugabaloo Shoes Addition: Students add two sets of numbers to find the sum.http://MGSEK.sheppardsoftware.com/mathgames/earlymath/bugabalooShoes.htm

Math Games: Fruit Shoot Addition: Students add two numbers together and then choose the correct sum.http://MGSEK.sheppardsoftware.com/mathgames/fruitshoot/fruitshoot_addition.htm



http://www.sheppardsoftware.com/mathgames/fruitshoot/fruitshoot_addition.htm

http://www.sheppardsoftware.com/mathgames/earlymath/bugabalooShoes.htm



PROBLEM TYPES

Join/Combine

Mouse had 6 cookies. Emma gave him 4 more. How many

cookies does Mouse have?

6 + 4 = ?

Mouse had 6 cookies. Emma gave her some more. Now Mouse has 10cookies. How many did Emma give her?

6 + ? = 10

Mouse had some cookies. Emma gave her 4 more.

Now Mouse has 10 cookies. How many

cookies did Mouse have to begin with?

? + 4 = 10

Separate/Decompose

Mouse had 9 cookies. He gave 7 to Emma. How many cookies does Mouse have left?

9 – 7 = ?

Mouse had 9 cookies. He gave some to Emma. Now he has 2 cookies left. How many did he give to Emma?

9 - ? = 2

Mouse had some cookies. He gave 7 to Emma. Now he has 2 cookies left. How many cookies did Mouse have to start with?

? – 7 = 2

Total Unknown Addend Unknown Both Addends Unknown1

Put Together / Take Apart2

Mouse has 3 chocolate chip cookies and 5 sugar cookies. How many cookies does he have?

3 + 5 = ?

Mrs. Haktak has 8 cookies. 3 are chocolate chip and the rest are sugar cookies. How many cookies are sugar cookies?

8 – 3 = ?

Mouse has 5 cookies. How many are sugar and how many are chocolate chip?5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

Difference Unknown Greater Unknown Less Unknown

Compare3

(“How many more?” version):Spiders have 8 legs. Mice have 4 legs. How many more legs do spiders have than mice? (“How many fewer?” version):Spiders have 8 legs. Mice have 4 legs. How many fewer legs do mice have than spiders?

(Version with “more”):Mice have 4 legs. Spiders have 4 more legs than mice. How many legs do spiders have? (Version with “fewer”):Mice have 4 fewer legs than spiders. Spiders have 8 legs. How many legs do mice have?

(Version with “more”):Spiders have 8 legs. They have 4 more legs than mice. How many legs do mice have? (Version with “fewer”):Mice have 4 fewer legs than spiders. Spiders have 8 legs. How many legs do mice have?

The tasks in this chart provide a variety of word problems using addition and subtraction for students to explore. It is not necessary for students to be able to solve ALL problem types in

order to meet the standard for solving word problems using objects or drawings (MGSEK.OA.2)





CONSTRUCTING TASK : Building Trains (Addition/Combining)Back To Task Table

Approximately 1 day and repeated through centers (adapted from Chris Confer’s “Snap It” found in Teaching Number Sense)







1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

BACKGROUND KNOWLEDGE “When parts of a set are known, addition is used to name the whole in terms of the parts.” (Van de Walle & Lovin 2006) Building Trains allows children to focus on a single number for the entire activity. It is important to give the children time to work on a single number (usually to 4 or 5, at first) throughout the activity, allowing them opportunities to explore through a variety of materials and methods of joining or separating. As their understanding of concepts develops, encourage students to expand their understanding with higher numbers. Allowing students multiple opportunities to participate in these types of activities gives them the chance to think about number relationships in a relaxed setting (Burns 2007; Van de Walle & Lovin 2006).

This activity reinforces the concept of addition (and the inverse, subtraction) through part-part-whole models. For students to see the two parts and the whole, the two parts must be kept as two





separate parts. For example, if using counters instead of connecting cubes, the two groups should be kept in separate piles or separated by color or using a part-part-whole mat.


ESSENTIAL QUESTIONS

What happens when I join quantities together?

MATERIALS

Connecting cubes (up to 10 per student) or other manipulatives (2-sided counters, shapes, dominoes, to name a few.)

Ten Black Dots (Donald Crews 1986) or similar counting book Building Trains recording sheet

GROUPING

Whole, Individual and/or Small group task

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Part 1 (Addition)Before reading the book, Ten Black Dots by Donald Crews, or a similar book involving addition (see literature list for suggested titles), ask students to turn to a partner and show them “10” using their “math hands” (notice students who can show ten immediately and those who may count each finger or count on from 5 fingers). As you read, stop when you come to each number and ask students how many more black dots you would need to get to ten. (For example, on the page that says “…Two dots can make the eyes of a fox or the eyes of keys that open locks.” Say, “I see two black dots, how many more dots are needed to make ten?”)

Part IIAfter reading, begin an investigation to explore number relationships within 5 (and later within 10). Give each student five connecting cubes. Explain that when you say, “Break”, that is their cue to break the train of cubes into two equal parts. When they have separated their train of cubes, they will need to notice how many are in each group and share the combination they made with the class. Model for them how to share the combination (“I had 5 cubes. As I break my train in half, I am left with a set of 3 and a set of 2. So, 2 and 3 is equal to 5.”). Practice the activity with the class and have a few students share their combinations. Once you see that students understand the activity, assign partners and give each set of partners a sum of cubes to work with according to their abilities (numbers within 5 or 10). Students will take turns breaking their trains in two parts and show them to their partners. The partner will then say the combination aloud and repeat the process after switching roles.

Have students share their thoughts about the activity they participated in today. Have them share about any patterns they noticed during the activity (“3 + 2 is the same as 2 + 3. So, it doesn’t





matter the order of the addends, the sum is the same.”) and allow them to share their thinking. For students that have difficulty writing their math thinking, allow them to dictate to you or someone who can scribe their thoughts.

Part III (Five Tower Game)“Five-Tower Game” (Burns 2007) Students will roll 2 dice, add the numbers that come up and build a tower with that many connecting cubes. Each partner repeats the process until five towers are made. Once the towers are made, the two students compare the number of cubes each person has. How do they know who has more? Less? In addition, questions such as the following can increase engagement in what seems to be a low level activity.

What is the greatest amount of cubes you could earn?What is the least amount of cubes you could earn?What is the difference in two towers? Can you use all your cubes and make 2 equal towers? Can you justify your thinking?


Can students explain their thinking when solving an addition problem? How many different combinations are students able to create when adding to five?


Does the order of the addends change the sum? Can you explain your thinking? How do you know when your answer makes sense?

DIFFERENTIATION

Extension “Two out of Three” Game (Activity 2.17, page 50. Van de Walle)

Intervention Allow students to work with numbers that are appropriate for their performance level. If

students are still counting using one-to-one correspondence, you may suggest that they use numbers less than five. For those that seem to quickly identify number relationships to five, suggest numbers to ten until they can quickly identify number relationships within ten.

Allow students who are ready to record addition sentences to describe how they broke apart their trains.


TECHNOLOGY

Math Games: Pop Up Addition: Students solve addition problems by adding two numbers together.http://MGSEK.sheppardsoftware.com/mathgames/popup/popup_addition.htm



http://www.sheppardsoftware.com/mathgames/popup/popup_addition.htm



Building Trains (Addition) Name:__________________________________

Is the same as += and

is equal to +and Is the same as

+ =+ =





CONSTRUCTING TASK : Building Trains (Subtraction)Back To Task Table

Approximately 1 day and repeated through centers (adapted from Chris Confer’s “Snap It” found in Teaching Number Sense)







BACKGROUND KNOWLEDGE “In part-part-whole model, when the whole and one of the parts are known, subtraction names the other part.” (Van de Walle & Lovin 2006) Even when there is a remove action, the situation in this next activity ends with two clearly distinct parts. It is important for you to have a discussion with your students, which includes showing them how an addition and subtraction sentence can be written for the same situation, to help them connect addition and subtraction. (More information can be found in Chapter 3 of Teaching Student-Centered Mathematics: Grades K-3, Van de Walle & Lovin 2006)


ESSENTIAL QUESTIONS

What happens when sets are joined or separated?





MATERIALS

Connecting cubes (up to 10 per student) or other manipulatives (2-sided counters, shapes, dominoes, to name a few.)

Ten Black Dots (Donald Crews 1986) or similar counting book Building Trains recording sheet

GROUPING


TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Review the activity of Breaking Trains from the previous lesson. Ask them to share what patterns they may have noticed while working with the same number. After several students have had an opportunity to share, explain that this time, students will break their train and hide one of the parts behind their back. Their partner will have to identify the part shown and then predict the quantity of cubes that are behind their back. Once a prediction has been made, the partner with the cubes reveals how many are hidden for the other partner to check his or her thinking. They should both attempt to write the addition and subtraction sentence that describes the situation.

Before beginning, model the activity for the group. You will be “Player A” and the class will act as “Player B”. Using a set of ten, break the train and hide one part behind your back. All students will use their math hands to predict how many are behind your back (Scan to see who has made a logical prediction and those whose predictions are illogical. Intervention may be needed for those that demonstrate a lack of number sense.). After all students make a prediction with their math hands, reveal the number of cubes behind your back. Next, ask a volunteer to show the class how to write an addition and subtraction number sentence that would describe the situation you modeled. Encourage them to share their math thinking. Repeat this process until you feel that students are ready to practice this same activity with a partner.

Once students are ready, assign each set of partners a number that is appropriate for their performance level and allow them to begin working. As you walk around and observe partners working together, stop and ask students some of the following questions (or similar questions to assess student learning)

(as students show one part of cubes and hide the other part) What goes with (known part) to make _ (known whole) __? Or _ (known part) _ and ? make _ (known whole) _?

What strategies did you use to find your answer?Have students share their math strategies they used to determine how many cubes were missing. Have them share any patterns they noticed during the activity and allow them to share their thinking. For students that have difficulty sharing their math thinking, allow them to show their thinking.






Can students explain their thinking when solving a subtraction problem?


How do you know when your answer makes sense?

DIFFERENTIATION

Extension “Grab Bag Subtraction”-adapted (Burns 2007) Children can work in pairs or

independently for this activity. Fill a paper bag with any set of tiles or cubes (to 5 or 10) and write the quantity of items in the bag on the front of the sack. Students will reach in the bag and take some out, showing how many were removed. Both students will predict how many they think are left inside of the bag. Then they will check their predictions and record the addition and subtraction number sentences.

Intervention Students may use a part-part-whole mat for this activity and cover an addend with a sheet

of paper. Make sure that students flip the mat so that the “whole” section is first, showing that one part is removed from the whole. An example of a part part whole mat can be seen below.

whole

part partBack to Intervention Table





Building Trains (Subtraction) Name:__________________________________

Is the same as -= and

is equal to -minus Is the same as

- =- =





PRACTICE TASK: Going Bananas Back To Task TableApproximately 1-2 days STANDARDS FOR MATHEMATICAL CONTENT









Students need to develop an understanding of part-whole relationships as they recognize that a set of objects (5) can be broken into less sub-sets (3 and 2) and still remain the total amount (5). In addition, this objective asks students to realize that a set of objects (5) can be broken in multiple ways (3 and 2; 4 and 1). Thus, when breaking apart a set (decompose), students use the understanding that a less set of objects exists within that greater set (inclusion).

Use of the recording sheet for this task is optional because students need to learn to draw a representation of what they are thinking for three reasons:SMP2. Reason abstractly and quantitatively.





SMP3. Construct viable arguments and critique the reasoning of others.SMP4. Model with mathematics.

By encouraging students to organize their work and draw representations, students can begin to “talk through” their process. This enables students the opportunity to attend to precision as they explain and reason mathematically. Using tables, numbers, pictures and words allows students to become more efficient when identifying patterns in numbers and making generalizations.

Commutative property might be identified during this task. Have students discuss whether or not to include commutative properties as possible combinations. Since there are no right or wrong answers, students should construct viable arguments and explain why their combination of bananas is correct. In addition, students must be willing to analyze and critique the reasoning of others.


ESSENTIAL QUESTIONS

How can I use different combinations of numbers to represent the same quantity?

MATERIALS

Paper Pencil Accessible manipulatives 2 bowls per group/student

GROUPING


TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

It is useful to think of problem-based lessons as consisting of three main parts: before, during, and after. If you allot time for each part it is quite easy to devote a full period to one seemingly simple problem. (Van de Walle p.15-19)

The Before Phase: get students mentally prepared for the task, be sure the task is understood, and be certain that you have clearly established expectations beyond simply getting an answer.

The During Phase: the most important thing here is let go! Give students a chance to work without your guidance. Give them an opportunity to use their ideas and not simply follow directions. It is also important to listen. Find out how different children or groups are thinking, what ideas they are using, and how they are approaching the problem. In this phase hints may be provided but not solutions, and students should be encouraged to test ideas.





The After Phase: this is often where some of the best learning takes place. During the after phase students share emerging ideas and the community of class learners is developed. This will not develop quickly or easily and will be developed over time.

(For a more detailed description of the three-phase problem-based learning model see Van de Walle Teaching Student-Centered Mathematics K-3 pp 15-19)

The task presented to students is as follows:

The monkeys loved to pick bananas. They loved to count their bananas and put them into baskets. If the monkeys picked 5 bananas one morning, what are all the possible ways they could put the bananas into baskets? Explain and justify.

Possible solutions for decomposing 5 (The concept of commutative property can be explored but only if it is discovered by students.

5 = 3 + 25 = 2 + 35 = 4 + 15 = 1 + 4 5 = 5 + 05 = 0 + 5


Can students explain their thinking when solving an addition or subtraction problem? How many different combinations are students able to create when adding to five?


Are there any more ways to compose or decompose the number 5? How do you know? Why did you decide to do it this way? Are you sure that you have found them all? Why do you think so? How do you know? Did you develop a shortcut to find your answers? Did you identify any patterns or rules? Can you explain them? What is the most number of baskets the monkeys could use? What is the fewest number of baskets the monkeys could use?

DIFFERENTIATION

Extension Students can be encouraged to find the solution for multiple quantities up to 10 bananas.

In addition, students can create a story problem using a quantity of 5. The standard addresses pairs of numbers to 5, teaching open ended word problems in a

context allow students to go above and beyond the expectation of the standards. Mathematics GSE Kindergarten Unit 5: Investigating Addition and Subtraction


All Rights Reserved



5 = 1 + 1 + 1 + 1 + 15 = 2 + 1 + 1 + 1 + 05 = 2 + 2 + 1 + 0 + 05 = 3 + 1 + 1+ 0 + 05 = 3 + 2+ 0 + 0 + 05 = 4 + 1+ 0 + 0 + 05 = 5 + 0 + 0 + 0 + 0

Intervention Allow students to work through the stages at a speed that is appropriate for their

performance level. Some students may need additional experiences acting out problems, using manipulatives, or drawing pictures.

Students can use cut out pictures of bananas and act out the story by placing the cut outs in bowls or pictures of bowls.

Start with a number less than 5 and allow students to build and generalize with the patterns that they are seeing. In addition, you could limit the number of baskets that a student uses.


TECHNOLOGY

Brainie: Students find different combinations to build a numberhttp://MGSEK.coolmath-games.com/0-brainie/index.html



http://www.coolmath-games.com/0-brainie/index.html



Going BananasThe monkeys loved to pick bananas and put them into two baskets. If the monkeys picked 5 bananas one morning, what are all the possible ways they could put their bananas into baskets? Explain and justify.





Going BananasThe monkeys loved to pick bananas and put them into two baskets. If the monkeys picked 10 bananas one morning, what are all the possible ways they could put their bananas into baskets? Explain and justify.









PRACTICE TASK: Tug-O-War Back To Task TableApproximately one day and repeated through centers STANDARDS FOR MATHEMATICAL CONTENT




1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.


Many children will use number lines (models) to solve story problems. The model is a thinking tool to help them both understand what is happening in the problem and a means of keeping track of the numbers and solving the problem. Tug-O-War is a task that helps students become familiar with adding and subtracting along number line. (Van de Walle p.72-72)


ESSENTIAL QUESTIONS

How can using benchmark numbers help me when adding or subtracting? How can I use models to represent addition and subtraction?

MATERIALS

1 counter Spinner Tug-O-War game board





GROUPING

Partners (2 players)


Place the counter at the number 5 on the number line.

Player 1 spins the spinner and moves the counter the correspoding number of spaces towards zero on the number line. (If the chip is on five and player 1 spins a 3 they move the chip 3 spaces to 2) Player 1 must identify and say the number of where the chip is on the number line. If player 1 is unable to correctly identify the correct location of the chip, it moves back to the previous location.

While the game is being played students should use a ten frame to model what is happening to the counter on the number line which in turn will help with strategy development. Although writing the equation is NOT a standard that is explicitly listed in Kindergarten, it is suggested that students be provided with lots of opportunities to record their thinking. Students can record the results of their turn in their journal.

Player 2 spins the spinner and moves the counter that many spaces towards 10. (If the chip is on 7 and player 2 spins a 3, they move the chip 3 spaces to 10) If player 2 is unable to identify the correct location of the chip, it moves back to the previous location.

If the chip reaches zero on the number line, player 1 wins. If the chip reaches 10 on the number line, player 2 wins.


How are students moving the counter on the number line? Are students counting by ones? Or are students using a new strategy? Do students use the benchmark numbers to count forwards or backwards to determine the

location of their counter?


Instead of counting 1 by 1, could you have located the counter a different way? Are you using any strategies to help you find where the counter should be placed? Did you use the benchmark numbers to locate the place on the number line? Where is the counter located on the number line right now? How do you know? Did you develop a shortcut to find your answers? Did you identify any patterns when playing the game?





DIFFERENTIATION

Extension 2 six-sided die could be used to extend the game. Have players roll the dice, determine

the answer and move the corresponding number of spaces.

Intervention The numerals can be written on the Tug-O-War game board to help assist students that

struggle with identifying the location on the number line.


TECHNOLOGY

Addition Games-Arrgh! Pirate Platform: Students add two addends to find a quantity while traveling through a pirate shiphttp://MGSEK.fun4thebrain.com/addition/arrrgh.html



http://www.fun4thebrain.com/addition/arrrgh.html



Tug-O-War

Materials1 counter and spinner numbered (1-3)Directions

1. Place the counter at the number 10 on the number line. 2. Player 1 spins the spinner and moves the counter the correspoding number of spaces towards zero on the number line. (If

the chip is on five and player 1 spins a 3 they move the chip 3 spaces to 5) Player one must identify and say the number of where the chip is on the number line. If player 1 is unable to correctly identify the correct location of the chip, it moves back to the previous location.

3. Player 2 spins the spinner and moves the counter that many spaces towards 10. (If the chip is on 7 and player 2 spins 2 they move the chip 2 places to 9) If player 2 is unable to correctly identify the correct location of the chip, it moves back to the previous location.

4. If the chip reaches zero on the number line, player 1 wins. If the chip reaches 10 on the number line, player 2 wins.



0

4 10

2 64 8 10

0 2 6 8



Spinner

1 2

3

Spinner

1 2

3





PRACTICE TASK: Race To 20 Revisited Back To Task TableApproximately one day and repeated through centers STANDARDS FOR MATHEMATICAL CONTENT






In this task students will roll the dice and make a 5. If a student rolls a 3 they would need 2 more to make a 5, so they move 2 spaces. Students will eventually roll a 6. This will provide an opportunity for class discussion and for students to engage in SMP3. In order to make a 5 from 6, one would need to be removed. Have students discuss how this could be represented in the game (move backwards one space).


ESSENTIAL QUESTIONS

How can using benchmark numbers help me when adding or subtracting?

MATERIALS

2 different colored counters 1 number cube (1-6) Race to 20 game board

GROUPING

Partners (2 players)






Each player places their counter in the starting square.

Player one rolls the dice and moves as many spaces needed to make a 5. Example: 3 rolled, move forward 2 places. 4 rolled move forward 1.

Players must correctly identify what number square they are on and how they know or return back to the previous square.

Players alternate turns until one player reaches 20.


Are students counting by ones? Or are students using a new strategy? Do students use the benchmark numbers to count forwards or backwards to determin the

location of their counter?


Instead of counting 1 by 1, could you have located the counter a different way? Are you using any strategies to help you find where the counter should be placed? Did you use benchmark numbers to locate the place on the board? Where is the counter located on the board right now? How do you know? Did you develop a shortcut to find your answers? Did you identify any patterns when playing the game?

DIFFERENTIATION

Extension (2) sixed-sided dice could be used to extend the game. Have players roll the dice,

determine the answer and move the corresponding number of spaces on the board.

Intervention While the game is being played students can use a five-frame to model what is happening

to the counter. This can also be beneficial for students that struggle to identify strategies.






Start 5Great Tire Change…

move another 5

10

Finish

20

15

Flat Tiremiss a turn



Race to 20Materials:2 different colored counters1 number cube (1-6)

Directions:-Each player places their counter in the starting square.-Player one rolls the dice and moves as many spaces needed to make a 5. Example: 3 rolled move forward 2 places. 4 rolled move forward 1.-players must correctly identify what number square they are on and how they know or return back to the previous square.-Players alternate turns until one player reaches 20.



CONSTRUCTING TASK: Shake and Spill Addition (5 Or 10) Back To Task Table

Approximately one day and repeated through centers (adapted from Marilyn Burns’ Shake and Spill activity found in About Teaching Mathematics: A K-8 Resource








BACKGROUND KNOWLEDGE “When parts of a set are known, addition is used to name the whole in terms of the parts.” (Van de Walle & Lovin 2006) Shake and Spill allows children to focus on a single number for the entire activity. It is important to give the children time to work on a single number (usually to 4 or 5, at first) throughout the activity, allowing them opportunities to explore through a variety of materials and methods of joining or separating. As their understanding of concepts develop, encourage students to extend their understanding with higher numbers. Allowing students multiple opportunities to participate in these types of activities gives them the chance to think about number relationships in a relaxed setting (Burns 2007; Van de Walle & Lovin 2006).





This activity reinforces the concept of addition (and the inverse, subtraction) through part-part-whole models. For students to see the two parts and the whole, the two parts must be kept as two separate parts.


ESSENTIAL QUESTIONS

Why is it important that I can build the number combinations for the number 5? 10?

MATERIALS

Two-color counters One small cup, per student (like a mouthwash cup) One dry-erase marker per student Shake and Spill recording sheet for addition Anno’s Counting House by Mitsumasa Anno, or another similar counting book

GROUPING

Whole, individual, and/or small group task


Begin this task by “reading” the math picture book, Anno’s Counting House (Anno 1982) or a similar book. As some of the children in the story move to the other house, ask questions such as, “Who’s gone? How many are gone? If ____ are gone, how many must be in the first house?” With rich conversation during this story, all combinations of ten will be discussed.

After reading, begin an investigation to explore number relationships within 5 (and later within 10). Give each student a cup of five double-sided counters. Ask students to swirl the cup of counters and then spill them on their table or work space. How many red counters do you see? How many yellow? Then show students how to record what they see using a number sentence. Allow them to write on their table, using their dry-erase marker, the same number sentence. Repeat this process until you see that a majority of your students are ready to practice on their own.

When showing students how to record a number sentence, make sure to identify all the parts: the addends, the equal sign (can be read, “the same as”), and the answer to an addition number sentence, or sum. Take care to ensure that students can differentiate between the term “sum” and the word “some”. Once students demonstrate an understanding of the activity, allow them to do the activity on their own. Assign each student a number of counters to work with according to his or her ability level (5 or 10). Each student will swirl and spill the counters in his cup, draw what they see, and write a number sentence that describes what is shown.





Possible solutions: 5= 0+5, 5+0=5, 4+1=5, 1+4=5, 3+2=5, 2+3=5 10+0=10, 0+10=10, 9+1=10, 1+9=10, 8+2=10, 2+8=10, 7+3=10, 3+7=10, 6+4=10, 4+6=10, 5+5=10

Possible questions that engage students: When recording the number of red sides and yellow sides each time you spill the counters, do you think you’ll get one result more often than others?If so, what will it be? How do you know?


How many number combinations were students able to make with 5 or 10? What did they notice about the combinations?


Does the order of the addends change the sum? Can you explain your thinking. How do you know when your answer makes sense?

DIFFERENTIATION

Extension Play, “Five-Frame Tell-About” (Van de Walle, page 46). This game could also be

changed to “Ten-Frame Tell-About”.

Intervention Allow students who have difficulty with organization to use a 5 or 10s frame. As they

spill the counters, have them fill up the frames with the counters and then record their findings.

Students may use a part-part-whole mat for this activity. An example of part-part-whole mat can be seen below.

Part wholePart






TECHNOLOGY

Math Games: Pop Up Addition: Students solve addition problems by adding two numbers together.

http://MGSEK.sheppardsoftware.com/mathgames/popup/popup_addition.htm






Shake and Spill: Addition

Number of Red

Counters+ Number of

Yellow Counters

=is the

same as:

Total Number

of Counters





CONSTRUCTING TASK: Shake and Spill Subtraction (5 Or 10) Back To Task Table

Approximately one day and repeated through centers (adapted from Marilyn Burns’ Shake and Spill activity found in About Teaching Mathematics: A K-8 Resource








BACKGROUND KNOWLEDGE “In part-part-whole model, when the whole and one of the parts are known, subtraction names the other part.” (Van de Walle & Lovin 2006) Even when there is a remove action, the situation in this next activity ends with two parts clearly distinct. It is important for you to have a discussion with your students, which includes showing them how an addition and subtraction sentence can be written for the same situation, to help them connect addition and subtraction. (More information can be found in Chapter 3 of Teaching Student-Centered Mathematics: Grades K-3, Van de Walle & Lovin 2006)






ESSENTIAL QUESTIONS

What happens when I decompose a quantity? How do you know when your answer makes sense?

MATERIALS

Two-color counters One small cup, per student (like a mouthwash cup) One dry-erase marker per student Shake and Spill recording sheet for subtraction Anno’s Counting House by Mitsumasa Anno, or another similar counting book

GROUPING

Whole, individual and/or small group task


Review the activity of Shake and Spill from the previous lesson. Ask students to share what patterns they may have noticed while working with the same number. After several students have had an opportunity to share, explain that this time, students will write the total number of counters they begin with and then subtract the number of yellow counters that are spilled.

Before beginning, model the activity for the group. You will be “Player A” and the class will act as “Player B”. Using a set of five or ten, spill the counters in the cup and identify how many yellow counters are shown. Then think aloud, “So, if I started off with 5 (or 10) counters, and there are _____ yellow spilled, how many red counters were spilled? How many more do I need to get to 5 (or 10)?” All students will use their math hands to predict how many red counters were spilled (scan to see who has made a logical prediction and those who’s prediction are illogical. Intervention may be needed for those that demonstrate a lack of number sense.). After all students make a prediction with their math hands, reveal the number of red counters that were spilled. Next, ask a volunteer to show the class how to write an addition and subtraction number sentence that would describe the situation you modeled. Encourage them to share their math thinking. Repeat this process until you feel that students are ready to practice this same activity independently (or with a partner).

Once students are ready, assign each student a number that is appropriate for their performance level and allow them to begin working. As you walk around to assess students, stop and ask the following questions (or those similar to assess student learning):

What number did you start off with? So, if there are ___ yellow counters, how many red counters do you think there are? How do you know without counting?






Can students explain the strategies they used to determine how many red counters were spilled?

What do the students notice about this version of this game that is alike or different from the addition version?


How do you know when your answer makes sense?

DIFFERENTIATION

Extension “Grab Bag Subtraction”-adapted (Burns 2007) Children can work in pairs or

independently for this activity. Fill a paper bag with any set of tiles or cubes (to 5 or 10) and write the quantity of items in the bag on the front of the sack. Students will reach in the bag and take some out, showing how many were removed. Both students will predict how many they think are left inside of the bag. Then they will check their predictions and record the addition and subtraction number sentences.

Intervention Students may use a part-part-whole mat for this activity. Make sure that students flip the

mat so that the “whole” section is first, showing that the one part is removed from the whole. An example of a part part whole mat can be seen below.

Who

le

part

part






Shake and Spill: Subtraction

Total Number of counters

- Number of Yellow

Counters

=is the

same as:

Number of Red

Counters





PRACTICE TASK: SUM-moning 5’s and 10’s Back To Task TableApproximately one day and repeated through centers









Students can develop and refine strategies as they hear other students' descriptions of their thinking about number combinations. For example, a student might compute 4 + 5 by counting on from 4: "...,5, 6, 7, 8, 9." But during a class discussion of solutions for this problem, she might hear another student's strategy, in which he uses knowledge about 5; namely, 3 and 2 make 5, and 5 more is 10. She may then be able to adapt and apply this strategy later when she computes 12 + 7 by saying, "12 and 3 make 15, and 4 more is 19" (adapted fromNCTM Principles and Standards, 2012).

For more information about common misconceptions, please refer to the unit overview.ESSENTIAL QUESTIONS

Why is it important that I can build the number combinations for the number 5? 10?





MATERIALS

1 SUM-moning FIVES game board, per pair of students 5 double-side counters, per student 2 dice, per pair of students five frame

GROUPING

Partner task


The first player rolls two dice and finds the sum.

After finding the sum of the two numbers, the student will decide what can be added to 5 to make the same sum. After a prediction has been made and explained, students model their explanation on the five frame board. If the student was correct, they place a double- sided counter over the equation that matches their prediction, and five frame explanation.

If the equation is not written, or if 5 cannot be used in an equation that is on the board, the player skips a turn to fill the board.

The two players alternate turns until the first player fills up his game board.

The person with the board filled first, wins!

Have students share with the whole group the strategies they used to identify how many more were needed to make a number (5 or 10). Were there situations in which they added more than two numbers or that they subtracted two numbers? If so, what were they? Challenge students to identify all the roll combinations where 5 or 10 can’t be made with 2 dice.


Do students use mathematical language to explain the strategies they used? Are students able to use various strategies to solve problems?


What strategies are you going to use to find the sum of two numbers? If you found sums higher than 5, what strategies did you use?

DIFFERENTIATION

Extension





Change the game board to SUM-moning TENS and follow the same procedures as listed above.

Teachers may also use activity 2.20 found on page 51 (Van de Walle & Lovin, 2006), “I Wish I Had”. After modeling this activity for students, have them work in pairs to play.

Intervention This game board can be changed to include only number combinations that are less than

5 or 10. For students who are not yet ready to work with sums beyond 5 or that are not yet proficient with using 5 as a benchmark.


TECHNOLOGY

Early Math: Bugabaloo Shoes Addition: Students add two sets of numbers to find the sum.http://MGSEK.sheppardsoftware.com/mathgames/earlymath/bugabalooShoes.htm



http://www.sheppardsoftware.com/mathgames/earlymath/bugabalooShoes.htm



5 55+1 5+15+2 5+25+3 5+35+4 5+4



SUM-moning 5’s



SUM-moning 10’s

10 1010+1 10+110+2 10+210+3 10+310+4 10+410+5 10+5





FAL: Decomposing Numbers Operations and Algebraic Thinking

Please use the following website to access the FAL: https://ccgpsmathematicsk-5.wikispaces.com/K-5+Formative+Assessment+Lessons+%28FALs%29





PRACTICE TASK : Bo Beep’s Domino Dilemma Back To Task TableApproximately 1-2 days









BACKGROUND KNOWLEDGE Numbers are related to each other through a variety of number relationships. The number 7, for example, is 3 more than 4, two less than 9, composed of 3 and 4 as well as 2 and 5, is three away from 10, and can be quickly recognized in several patterned arrangements of dots. These ideas further extend to an understanding of 17, 99, and beyond. Number concepts are intimately tied to the world around us. Application of number relationships to the real world marks the beginning of making sense of the world in a mathematical manner (Van de Walle, 2010).






ESSENTIAL QUESTIONS


MATERIALS

Recording sheet (optional) Accessible manipulatives or dominos

GROUPING

Whole, individual and/or small group task


Part IIf students have never been introduced to the game of dominoes, allow students to explore and play a couple of games before the task begins. Have students discuss the strategies they used during the game. (SMP 1-8)

Part II MemoryAll the dominoes are placed face down. In partners, students take turns flipping over two dominoes where the total number of pips on the dominoes match. Students must find two different dominoes that have the same total (Example: one domino could have 3&2 and the matching domino could have 4&1). Once all dominoes have been matched up and no more matches are possible, the player with the most pairs of dominoes wins. (SMP 1-8) Part IIIMatch players by ability level and have them take turns flipping over a domino. The first player to call out the total sum of pips on the domino wins. However, in order for the player to collect the domino, they must describe the strategy they used to their partner. (Example: 5&2 Student says “7, and I know that because 2 more than 5 is 7. If the domino was 3&0 the student might say “3, and I know that because any number plus zero is the number”).

After all the dominoes have been played, the player with the most dominoes wins. (SMP 1-8)

Part IV (to 5)Introduce students to the task of 5 pips. Begin by discussing the vocabulary in the task and check for understanding.

Little Bo Peep loved to play dominoes. She had one domino in her pocket. The sum of the pips on her domino was 5. What could the domino in Bo Peep’s Pocket look like?

Comment: there are more than enough squares provided for the task to 5. This is used to make students justify that they have found all of the possible combinations. Many times students will





know they are complete when all the squares have been “filled up”, so by providing more than enough squares, this cannot occur. (SMP 1-8)

Part V (to 10)Introduce students to the task of 10 pips. Begin by discussing the vocabulary in the task and check for understanding.

Little Bo Peep loved to play dominoes. One day she had 2 dominoes in her pocket and the sum of the pips on her two dominos was 10. What could have been the two dominoes in Bo Peep’s Pocket?

Comment: there are not enough squares provided for this task. Any additional combinations to ten that students identify should be recorded on the back of their sheet.. (SMP 1-8)


How many number combinations were students able to make with 5 or 10? What did they notice about the combinations?


What ways are the different combinations you found? Why did you decide to do it this way?

Are you sure that you have found them all? Why do you think so? Did you develop a shortcut to find your answers? Did you identify any patterns or rules?

DIFFERENTIATION

Extension Have the students identify which number of pips is the most frequent amongst the

dominos. Have students make a make a conjecture (draw a conclusion) explaining why the number

is most frequent.

Intervention Use a blank domino mat or folded piece of paper and give the student 5 or 10 counters.

Have students create the domino using the template and counters. Students can then transfer model to the recording sheet.






Bo Peep’s Domino DilemmaLittle Bo Peep loved to play dominoes. One day she had a domino in her pocket and the sum of the pips on her domino was 5. What could the domino in Bo Peep’s pocket look like?

Domino Total

_______ _______

_______ _______

_______ _______





Domino Total

_____ _____

_____ _____

_____ _____

_____ _____Domino Total

_____ _____

_____ _____

_____ _____

_____ _____





Bo Peep’s Domino DilemmaLittle Bo Peep loved to play dominoes. One day, she had 2 dominoes in her pocket and the sum of the pips on her two dominoes was 10. What could have been the two dominoes in Bo Peep’s Pocket?

Domino #1 Domino #2 Total

_______ _______ _______ _______

_______ _______ ______ ______

_______ _______ ______ ______





Domino #1 Domino #2 Total

_______ _______ _______ _______

_______ _______ ______ ______

_______ _______ ______ ______

_______ _______ ______ ______





PERFORMANCE TASK: The Bike Shop Back To Task TableApproximately 1-2 Days










In developing the meaning of operations, teachers should ensure that students repeatedly encounter situations in which the same numbers appear in different contexts. For example, the numbers 3, 2, and 5 may appear in problem-solving situations that could be represented by 2 + 3, or 3 + 2, or 5 – 3, or 5 – 2. Although different students may initially use quite different ways of thinking to solve problems, teachers should help students recognize that solving one kind of problem is related to solving another kind. Recognizing the inverse relationship between addition and subtraction can allow students to be flexible in using strategies to solve problems (NCTM Principles and Standards, 2012).






ESSENTIAL QUESTIONS


MATERIALS

The Bike Shop Recording sheet Bicycle cut-outs (if necessary)

GROUPING

Whole group, individual and/or small group task


Part IBegin by discuss bikes and bike shops with students. Allow students to share any experiences they have had with bikes. Specifically, discuss different types of bikes and what makes them different (focus on the wheels in particular). Ask the students the question “if I have 3 wheels sitting at home in my garage, what types of bikes could be in my garage?” Students will immediately say a tricycle but have them mentally explore possible combinations of bikes (unicycle and bike, for example). (SMP 1,2,3,6,7)

Part IIGive each student a workmat and picture sheet. Present students with the following task being sure to go over terms and understanding. For example, many students will know what a display case is, but not the formal name for it.

The Bike Store sold unicycles, bicycles, tricycles and go-carts. The owner of the store could only fit a total of 5 wheels in the display case at the front of the store. What are the different combinations of bikes and go-carts that could be put on display in the front of the store? Explain and justify your mathematical thinking.

(SMP 1-8)

Part IIIBegin this part by allowing students to explore the possible combinations of bike wheels for number less than 10. Ask the students “if there were 5 wheels and 3 seats in the front display case, what could be in the display?” Students will struggle more with this concept because of the addition of a specific number of seats. As students work through this task watch for students that have more or less than 3 types of bikes. This would indicate that students don’t understand the purpose of the seats and how the number must remain constant. After students have solved this problem present the following task:

The Bike Store sold unicycles, bicycles, tricycles and go-carts. In the front display case they are 10 wheels and 6 seats. What types of bikes are on display in the front of the Bike Shop? Explain and justify your mathematical thinking. (SMP 1-8)






Can students explain the different combinations they found with each scenario? Were students able to use different strategies when solving problems?

FORMATIVE ASSESSMENT QUESTIONS:

Are there more was to decompose the number 5 or 10? Why did you decide to do it his way?

Are you sure that you have found them all? Why do you think so? Did you develop a strategy to find your answers? Did you identify any patterns or rules? Are there any paired combinations to 5/10 that you were unable to model? If so, share

your thinking.

DIFFERENTIATION

Extension Have students create a combination of bike wheels that are equal to 10. Partners play and

try to identify the combination used by their partner.

Intervention Use clip art of unicycles, bicycles, tricycles, and go carts for students to manipulate and

exchange one type of bike for another. For Part III (specific combination to 10), have students find any combinations to 10 that

are possible and remove the restriction of 6 seats.


TECHNOLOGY







The Bike ShopThe Bike Store sold unicycles, bicycles, tricycles and go-carts. The owner of the store could only fit a total of 5 wheels in the display case at the front of the store. What are the different combinations of bikes and go-carts that could be put on display in the front of the store? Explain and justify your mathematical thinking.





The Bike ShopThe Bike Store sold unicycles, bicycles, tricycles and go-carts. In the front display case they are 10 wheels and 6 seats. What types of bikes are on display in the front of the Bike Shop? Explain and justify your mathematical thinking.









CONSTRUCTING TASK: Calling Out Reds! Back To Task TableApproximately one day and repeated through centers (adapted from the Marilyn Burns activity, How Many Reds?)










“Children need many opportunities to identify quantities, see relationships between numbers, and learn about the operations of addition and subtraction. When developing beginning concepts of numbers, children benefit from exploring concrete materials and relating numbers to problem situations. They also benefit from talking about their ideas and hearing how other children think. The learning activities should be varied in their contexts and involve the children directly in thinking, reasoning, and solving problems.” (Burns 2007)






ESSENTIAL QUESTIONS

Does the order of addends change the sum? How do you know when your answer makes sense?

MATERIALS

Calling Out Reds? recording sheets Two different colored dice, per set of partners Anno’s Counting House by Anno Mitsumasa

GROUPING



Refer to the book, Anno’s Counting House (Anno 1982) read in a previous lesson or “re-read” the book but pose different questions, such as, “What if the children moved in pairs instead of one at a time? What if there were three houses? What if there were more children? What else could be in the house to count? How many rooms, pictures, windows? What about your house? What about two classrooms or two buses instead of houses?” (Van de Walle & Lovin 2007) Have the children use counters to work out the story according to the new questions you pose.

Select two students to model this game for the class (or select two pairs of students and divide your class in half, so that a more intimate setting is provided for students to observe each pair of students playing the game. You could facilitate one group, your paraprofessional and/or collaborative teacher could facilitate the other.). The children will need twenty playing cards in all, consisting of ace through 10 of a red suit and black suit along with a recording sheet.

One child will mix up the cards, and deal them out, facedown, so that each child has 10 cards. Each student will count the number of red cards they have and write it on their recording sheet. Then, they will predict how many red cards their partner has so that they can make ten (show the students how to write the addition sentence). After each partner has written down the number of red cards they have and the number of red cards they predict their partner has to equal ten, they will check their work by each revealing the number of red cards they have. Have the pairs repeat this process until you determine that the whole group is ready to play. Once students are ready to play have students work with their math partner for this activity.

Invite each pair of partners to come to the whole group with their recording sheets. Make a class graph of the addends the children recorded. Analyze the graph and have students speculate about why some pairs of addends come up more often that others (Burns 2007).

Teacher Friendly Tip: Copy the addition and subtraction recording form back to back, laminate them (or use a sheet protector), and you have recording sheets that will last you a lifetime without making multiple copies! You will only need one class set. Students will use overhead





markers to record their answers. If you have a student on whom you are collecting data, make a copy before they erase their work and you have documentation of progress.(SMP 1-8)


Do students notice a pattern? Can students identify what pair of addends they have the most of? The least of? Are students able to determine how many more they will need to make ten, without

counting?


Are there more ways to set the bikes in the display case? Why did you decide to do it his way? Are you sure that you have found them all? Why do you think so? Did you develop a shortcut to find your answers? Did you identify any patterns or rules?

DIFFERENTIATION

Extension Students can play Make 10 on the Ten-Frame (Van de Walle, Activity 4.8; Van de Walle

& Lovin 2006) This activity can be adapted by changing it to “Make 5 on the Five-Frame” to keep the sum within ten. Use fact cards that have sums less than ten and give each student two five-frames to use to model each number on the fact card. Then, students should decide the easiest way to show what the total is, without counting. Take note as students explain their thinking of which strategies they are using-are they efficient strategies?

Intervention As students count their red cards, they could fill up a ten-frame with the corresponding

number of counters. They would record that number on their recording sheet and then identify how many more are needed to make ten. The student should then check their predictions by actually counting out the number of red cards their partner has.






Calling Out Reds?Name:_________________________

Number of Red Cards

I have+ How many

red cards my partner

has

= Total Number of Red cards





PRACTICE TASK: Dropping Pennies Back To Task TableApproximately 1-2 days










Numbers are related to each other through a variety of number relationships. The number 7, for example, is 3 more than 4, two less than 9, composed of 3 and 4 as well as 2 and 5, is three away from 10, and can be quickly recognized in several patterned arrangements of dots. These ideas further extend to an understanding of 17, 99, and beyond. Number concepts are intimately tied to the world around us. Application of number relationships to the real world marks the beginning of making sense of the world in a mathematical manner (Van de Walle, 2010).


ESSENTIAL QUESTIONS

How can I use different combinations of numbers to represent the same quantity?Mathematics GSE Kindergarten Unit 5: Investigating Addition and Subtraction


All Rights Reserved



MATERIALS

Pencil Recording sheet Pennies

GROUPING

Whole group, individual and /or small group


Part IReview or introduce a penny, and discuss how both sides of the penny are different. Be sure that students understand which side is heads on the coin and which side is tails. Draw a chart that is similar to the Dropping Pennies recording sheet. Give each student 3 coins and have them explore all the ways that the coins can drop and record the possible combinations. DO NOT complete the chart. Take only a few combinations and then present students with Part II. (SMP 1-8)

Part IIComment: there are more squares provided on the recording sheet than actual combinations. This is so that students can justify their answer.

Sam dropped 5 pennies on the ground. Some were heads up and some were tails up. How might the pennies have fallen? Show all the ways the coins could have landed on the ground.

In closing have students share the combinations they found and any strategies they used to solve the problem.(SMP 1-8)


Do students notice a pattern? Can students identify what pair of addends they have the most of? The least of? Are students able to determine how many more they will need to make five, without

counting?


Are there any more was to decompose the number 5? Why did you decide to do it his way? Are you sure that you have found them all? Why do you think so? Did you develop a strategy to find your answers? Did you identify any patterns or rules?





DIFFERENTIATION

Extension Instead of giving the student 5 pennies, they could use 5 coins that have total value of 20

cents or less. At the end of the task, students would be asked to add up their coins and justify their total value using numerals, pictures, and words. This could be recorded on the back side of the page.

Intervention Have the students act out the problem by dropping pennies on the ground and recording

the result. No result can be repeated. Because all combinations may never be the result of acting out this task, have the student determine the missing solution(s) through questioning.


TECHNOLOGY

Addition Games-Arrgh! Pirate Platform: Students add two addends to find a quantity while traveling through a pirate ship.http://MGSEK.fun4thebrain.com/addition/arrrgh.html



http://www.fun4thebrain.com/addition/arrrgh.html



Dropping PenniesSam dropped 5 pennies on the ground. Some were headsup and some were tails up. How might the pennies have

fallen? Show all the ways the coins have landed on the ground.Heads Tails Total

Dropping PenniesSam dropped 10 pennies on the ground. Some were headsup and some were tails up. How might the pennies have

fallen? Show all the ways the coins have landed on the ground.Heads Tails Total









PRACTICE TASK: Make 5 (Or 10) Go Fish! Back To Task TableApproximately one day and repeated through centers










“A mental computation strategy is simply an invented strategy that is done mentally. What may be a mental strategy for one student may require written support by another…As your students become more adept, they can and should be challenged from time to time to do appropriate computations mentally. Do not expect the same skills of all students.” (Page 161: Van de Walle & Loving, 2006)






ESSENTIAL QUESTIONS

How can I solve and represent problem situations using objects, pictures, words and numbers?

MATERIALS

1 deck of playing cards for each pair of students, face cards removed (ten frame cards will also work for this game)

GROUPING

Partners and/or small group task


Part IEngage students in a discussion about the possible number combinations to make 10. After students have explored possible combinations, introduce them to Make 10 Go Fish.

How to play: The object is to get two cards that total 10.

Each player is dealt five cards. The rest of the cards are placed down in the center of the table.

If you have any pairs of cards that total 10, put them down in front of you and replace those cards with cards from the deck.

Take turns. On your turn, ask the other player for a card that will go with a card in your hand to make 10.

If you get a card that makes 10, put the pair of cards down. Take another card from the deck. Your turn is over.

If you do not get a card that makes 10, take the top card from the deck. Your turn is over. (Example: Player 1 “Do you have a 2 in your hand?” If player 2 has a 2 they give it to player 1. If they do not have a 2 they say “Go Fish!”)

If the card you take from the deck makes 10 with a card in your hand, put the pair down and take another card. Your turn is over.





Practice Task: YOUNG FARMER MACDONALD Approximately 1-2 days Back To Task Table STANDARDS FOR MATHEMATICAL CONTENT







1. Make sense of problems and persevere in solving them. Students are able to compose and decompose numbers while solving problems involving addition and subtraction. 2. Reason abstractly and quantitatively. Students begin to draw pictures, manipulate objects, use diagrams or charts, etc. to express quantitative ideas such as a joining situation or separating situations. 3. Construct viable arguments and critique the reasoning of others. Students begin to clearly explain their thinking using mathematical language when composing and decomposing numbers. (Verbal and/or Written) 4. Model with mathematics. Students will begin to apply their mathematical thinking to real- world situations when given an addition or subtraction word problem. 5. Use appropriate tools strategically. Students will use manipulatives such as counting bears, cubes and number lines to model addition and subtraction problems. 6. Attend to precision. Students attend to the language of real-world situations to make sense of addition and subtraction problems. 7. Look for and make use of structure. Students begin to look for patterns and structure in the number system while exploring part-whole relationships using manipulatives. 8. Look for and express regularity in repeated reasoning. Students begin to recognize and use multiple strategies when combining and decomposing sets of numbers.






In developing the meaning of addition and subtraction with whole numbers, students should also encounter the properties of operations, such as the commutativity and associativity of addition. Although some students discover and use properties of operations naturally, teachers can bring these properties to the forefront through class discussions (NCTM Principles and Standards, 2012).

ESSENTIAL QUESTIONS

What happens when I decompose a quantity? How can I use different combinations of numbers to represent the same quantity?

MATERIALS

Recording Sheet Accessible manipulatives Pictures of barn animals or toy animals (not included)

GROUPING

Whole/individual/small group task


Part IEngage students in a class discussion about the farm and possible animals that could be seen if they visited the farm. When discussing the animals, have students pay particular attention to the number of legs on each animal. Show the students the picture of the pigs’ feet. Give students think time and allow them to share their thoughts of what animal the picture could be showing and why there are only 10 feet in view with the person they were sitting next to. Question the students asking, what animal do those feet belong to? How many feet does one pig have? How many pigs could be in the picture? This task will really make students make sense of the problem. As students share their thoughts, the teacher records them on the board or chart paper so that the rest of the class can justify that their thoughts are accurate.

After students have been able to share their thoughts about the picture, reveal the rest of the picture showing the whole picture of the three pigs.





Introduce students to the following task and check for student understanding of what is being asked of them.

Farmer MacDonald put all of her animals into different barns each night. Each barn could have no more than 10 feet. What animals did Farmer MacDonald put in her barn to equal ten feet? Find the different combinations of farm animals that equal 10 feet. Be sure to explain and justify your combinations.

Part II (What’s On My Farm)Have students create a specific combination of animals on the farm and play What’s On My Farm? Player 1 arranges a combination of animals and tells their partner how many legs and eyes they have on their farm in addition to the types of animals used.

Example: The combination of animals on my farm is one goat and a chicken so I would tell my partner “I have 6 legs and 4 eyes on my farm. What do I have on my farm?”

Example: The combination of animals on my farm is three goats and two chickens so I would tell my partner “I have 16 legs and 10 eyes on my farm. What do I have on my farm?” As students work through the extension ask them if they notice any patterns.

Comment: Students will know how many animals there are on the farm but looking at the number of eyes. (Example: if there were 6 eyes, there are 3 animals) This strategy should be identified by the students and not teacher prompted.


Are there any more ways to decompose the number 5? How do you know? Why did you decide to do it this way? Are you sure that you have found all of the ways? Why do you think so? How do you

know? Did you develop a shortcut to find your answers? Did you identify any patterns or rules? Explain!

DIFFERENTIATIONMathematics GSE Kindergarten Unit 5: Investigating Addition and Subtraction


All Rights Reserved



Extension Give students 3 numbers (1, 5, and 10). Ask students to make the number 20 using only

the given 3 numbers. Some possible combinations could be 10+10 or 10+5+1+1+1+1+1.

Intervention Print out pictures of farm animals or use animal figurines to manipulate while trying find

out the correct combination of farm animals. You could also give students a specific number of animals that were in the barn which would limit the possibility of combinations.






Young Farmer MacDonaldFarmer MacDonald needed to put all of her animals into different barns each night. Each barn could have no more than 10 feet. What animals did Farmer MacDonald put in her barn to equal ten feet?

Find the different combinations of farm animals that equal 10 feet. Be sure to explain and justify your combinations.









PRACTICE TASK: Four Sums in a Row Back To Task TableApproximately one day and repeated through centers STANDARDS FOR MATHEMATICAL CONTENT







1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.


As your students play this game, encourage them to use what they know from previous activities about number combinations to 5 and 10, and also what they know about using 5 and 10 as benchmark numbers. Young children often initially compute by using objects and counting; however, prekindergarten through 2nd grade teachers need to encourage them to shift, over time, to solving many computation problems mentally or with paper and pencil to record their thinking. Students should develop strategies for knowing basic number combinations (the single-digit addition pairs and their counterparts for subtraction) that build on their thinking about, and understanding of, numbers. Fluency with basic addition and subtraction number combinations is a goal for the pre-K–2 years (NCTM Principles and Standards, 2012).




4



ESSENTIAL QUESTIONS

Does the order of addends change the sum?

MATERIALS

1 Four Sums in a Row game board, per pair of students 2 clothespins 24 double-sided counters, per set of partners

GROUPING

Partner task


The first player places the 2 clothespins at the bottom of the game board and covers two addends. Player combines the addends and places a yellow marker over the sum.

The other player moves only one of the clothespins to a new addend and covers the sum with a red marker. Both clothespins can cover the same addend.

Play continues with pairs alternating turns, moving one neutral counter each turn until one player has four markers in a row (horizontally, vertically, or diagonally)

Have students share with the whole group the strategies they used to identify how many more were needed to make a number (5 or 10). Were there situations in which they added more than two numbers or that they subtracted two numbers? If so, what were they?(SMP1-8)


Are students able to develop a strategy to combine two addends and determine the sum? Able to combine two addends with a sum greater than 10?

Are students able to describe the strategies they used when combining addends using mathematical language?


What strategies are you going to use? What strategies did you use to get four sums in a row? If you found sums 11 and/or 12, what strategies did you use?

DIFFERENTIATIONMathematics GSE Kindergarten Unit 5: Investigating Addition and Subtraction


All Rights Reserved



Extension Activity 2.20, page 51 (Van de Walle & Lovin, 2006), “I Wish I Had”. After modeling

this activity for students, have them work in pairs to play.

Intervention This game board can be changed to include only number combinations to 5 or 10 for

students who are not yet ready to work with sums beyond 10, or are not yet proficient with using 5 as a benchmark.


TECHNOLOGY







Four Sums in a Row

3 10 12

9 6

9 2 5 3 73 8 7 0 57 4 8 6 9





5 12 1 2 40 1 2 3 4 5 6 7 8 9





Four Sums in a RowI kept the number...

I used the number…

Total I kept the number...

I made the number…

Total





CULMINATING TASK: The Candy Store Back To Task TableApproximately 1-2 Days










In developing the meaning of addition and subtraction with whole numbers, students should also encounter the properties of operations, such as the commutativity and associativity of addition. Although some students discover and use properties of operations naturally, teachers can bring these properties to the forefront through class discussions (NCTM Principles and Standards, 2012).


ESSENTIAL QUESTIONS






How can I represent problems using objects, pictures, and numbers?

MATERIALS

The Penny Pot by Stuart Murphy The Candy Store recording sheet and cut out candy (if needed)

GROUPING



Begin by reading Penny Pot (or a similar book) to the class and introduce the task. In this task two sisters go to the candy store and their mother has given them a nickel to spend. The sisters spend the whole nickel and each of them buys a piece of candy. Students must determine all of the possible combinations of candy they could buy. Be sure to discuss vocabulary and clarify any misconceptions students may have with the task.

Emma and Audrey went to the candy store and they had 1 nickel to spend between the two of them. If they spent the whole nickel and each got 1 item from the candy store, what are some of the items they could have bought?(SMP 1-8)


Do students understand that 1 nickel is the same as 5 cents? Are students able to decompose 5 in more than one way? Are students able to describe the strategies they used to solve the problem in

mathematical language?


Are there more was to decompose the number 5? Why did you decide to do it his way? Are you sure that you have found them all? Why do you think so? Did you develop a shortcut to find your answers? Did you identify any patterns or rules?

DIFFERENTIATION

Extension As an extension to this activity the amount of money and/or candy purchased could be

increased. In addition, students could find all the possible combinations of candy that could be bought using the entire nickel. In this extension more than two items can be bought. Example: 5 suckers





1 lollipop and 3 suckers2 lollipops and 1 sucker and 3 apples

Some students may realize that they can have as many apples as they want and the amount of money spent does not change the total. The zero property can be discussed ONLY IF IT IS IDENTIFIED BY THE STUDENTS.

Intervention Allow students to work through the stages at a speed that is appropriate for their

abilities. Some students may need additional experiences acting out problems, using manipulatives, or drawing pictures.

Students can use cut out pictures of candy and physically place them in combination pairs that make 5.

Students that have difficultly fixing a quantity to the candy can have the cost of each item written on the picture.


TECHNOLOGY

Math Games: Pop Up Addition: Students solve addition problems by adding two numbers together.http://MGSEK.sheppardsoftware.com/mathgames/popup/popup_addition.htm






The Candy StoreEmma and Audrey went to the candy store and they had 1 nickel to spend between the two of them. If they spent the whole nickel and each got 1 item from the candy store, what are some of the items they could have bought?

2 cents 4 cents 3 cents 5 cents 1 cent FreeLollipop Chocolate Bar Gum Milkshake Sucker Apple





The Candy StoreEmma and Audrey went to the candy store and they had 2 nickels to spend between them. If they spent both nickels and each got only 1 item from the candy store, what are some of the items they could have bought?

FreeApple

1 centsucker

2 centslollipop

3 centsgum

4 centsChocolat

e bar

5 centsmilkshak

e6 cents

chips7 cents

pop

8 centsjellybean

s9 centspopcorn

10 centsHotdog& pop





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Student Work Samples





KindergartenUnit 4 Investigating Addition and SubtractionTask: Dropping pennies(Instead of pennies we used 2 colored manipulatives)




GSE Investigating Addition and Subtraction Unit 5MGSEK.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects

or drawings, and record each decomposition by a drawing or equation (e.g., 5=2+3 and 5=4+1)











Documents

Type Grade Here - Wikispacesccgpsmathematicsk-5.wikispaces.com/file/view/K-Mat… · Web viewplease note that it is not until First Grade when “Understand the meaning of the equal